Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. $$ \operatorname{comp}_{b} \mathbf{c} $$

Step-by-Step Solution

Answer

The component of \( \mathbf{c} \) onto \( \mathbf{b} \) is 3.4.

Step 1: Understand the Concept

The problem asks for the component of vector \( \mathbf{c} \) in the direction of vector \( \mathbf{b} \). This is known as the scalar projection or component of \( \mathbf{c} \) onto \( \mathbf{b} \). The formula for this is given by: \( \operatorname{comp}_{\mathbf{b}} \mathbf{c} = \frac{\mathbf{c} \cdot \mathbf{b}}{\|\mathbf{b}\|} \).

Step 2: Calculate the Dot Product \( \mathbf{c} \cdot \mathbf{b} \)

Compute the dot product \( \mathbf{c} \cdot \mathbf{b} \) using the vectors \( \mathbf{c} = \langle -1, 5 \rangle \) and \( \mathbf{b} = \langle 3, 4 \rangle \). \[ \mathbf{c} \cdot \mathbf{b} = (-1)(3) + (5)(4) = -3 + 20 = 17 \]

Step 3: Find the Magnitude \( \|\mathbf{b}\| \)

Calculate the magnitude of \( \mathbf{b} \) using the formula for the magnitude of a vector: \( \|\mathbf{b}\| = \sqrt{3^2 + 4^2} \). \[ \|\mathbf{b}\| = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Step 4: Compute \( \operatorname{comp}_{\mathbf{b}} \mathbf{c} \)

Substitute the values found into the formula for the component of \( \mathbf{c} \) onto \( \mathbf{b} \):\[ \operatorname{comp}_{\mathbf{b}} \mathbf{c} = \frac{17}{5} = 3.4 \]

Key Concepts

Dot Product

To understand vector projection, it's important to first grasp the concept of the dot product. The dot product is a mathematical operation that takes two equal-length sequences of numbers and returns a single number. For two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their dot product is calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \). This operation tells us how much of one vector goes in the direction of another. It is essential in finding projections, angles between vectors, and in various applications in physics and engineering.

• The result of the dot product is a scalar, not a vector.
• If the dot product is zero, the vectors are orthogonal (perpendicular).

Understanding the dot product is critical as it forms the basis for finding the scalar projection of one vector onto another.

Magnitude of a Vector

The magnitude of a vector, also known as its length or norm, is a measure of how long the vector is. For a vector \( \mathbf{v} = \langle v_1, v_2 \rangle \), the magnitude is computed using the formula \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} \). This value is always non-negative and helps us understand the scale of the vector.

• The magnitude gives the distance from the origin to the point represented by the vector in a coordinate system.
• Vectors with a magnitude of 1 are called unit vectors.

In our solution, we used it to find the scalar projection by determining how much of one vector extends in the other vector's direction.

Scalar Projection

Scalar projection is the process of finding the length of the shadow of one vector on another vector. Mathematically, it is the component of one vector in the direction of another. For vectors \( \mathbf{a} \) and \( \mathbf{b} \), the scalar projection of \( \mathbf{a} \) onto \( \mathbf{b} \) is given by the formula:
\[ \operatorname{comp}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{b}\|} \]
This action results in a scalar, representing the effective length of \( \mathbf{a} \) in the direction of \( \mathbf{b} \).

• The scalar projection helps determine how much of one vector lies in parallel with another.
• The formula utilizes both the dot product and the magnitude of the vector.

Understanding scalar projection allows us to see relationships between vectors in a more quantified manner, especially when the vectors have different directions and magnitudes.

Vectors

Vectors are fundamental objects used in mathematics that have both magnitude and direction. They are represented as arrows pointing from one point to another or in coordinate form as \( \langle v_1, v_2 \rangle \) in two-dimensional space. Vectors are used not only in pure mathematics but also in physics for representing quantities like force and velocity.

• Vectors are often classified by their dimension; the simplest, are 2D and 3D vectors.
• Operations on vectors include addition, subtraction, scaling, and dot product.

Understanding vectors and their behavior is essential in a variety of fields, as they provide a way to represent and compute with quantities that have both direction and size.